The volume of the solid generated by revolving the region about the given line. above the curve y=3 below the curve y=sqrt of 3x on the left by the y-axis the line y=3
To find the volume of the solid generated by revolving the region about a line, we can use the method of cylindrical shells.
Step 1: Sketch the given region.
Start by sketching the region bounded by the curves y = 3, y = √(3x), and the y-axis. This will help you visualize the solid we are trying to calculate the volume of.
Step 2: Identify the axis of rotation.
In this case, the region is being revolved about the line y = 3. This means the axis of rotation is a horizontal line.
Step 3: Determine the limits of integration.
To set up the integral that calculates the volume, we need to identify the limits of integration. In this case, we will integrate with respect to y, so we need to express the x-values in terms of y. Let's start with the curve y = √(3x):
y = √(3x)
Squaring both sides:
y^2 = 3x
Dividing both sides by 3:
x = y^2/3
Since we're revolving the region around the y-axis, the "left" side of the region corresponds to the x-value of 0.
Thus, the limits of integration for y will be from y = 0 to y = 3.
Step 4: Find the radius and height.
To set up the integral using cylindrical shells, we need to express the radius and height of each cylindrical shell in terms of y.
The radius of each shell is the distance from the y-axis to the curve y = √(3x), which is x. Substituting x = y^2/3, we get the radius r = y^2/3.
The height of each shell is the difference between the upper and lower curves evaluated at y. In this case, the upper curve is y = 3 and the lower curve is y = √(3x). So the height h = 3 - √(3x). Substituting x = y^2/3, we get h = 3 - √(3(y^2/3)) = 3 - √y^2 = 3 - y.
Step 5: Set up and evaluate the integral.
The volume of each cylindrical shell can be expressed as V = 2πrh, where r is the radius and h is the height.
Therefore, the integral for the volume is:
V = ∫[0,3] 2π(y^2/3)(3 - y) dy
Integrating this expression with respect to y over the given limits [0,3] will give us the volume of the solid generated by the revolution.
After evaluating the integral, you will have the volume of the solid.